Properties

Label 1176.2.u.b.1097.8
Level $1176$
Weight $2$
Character 1176.1097
Analytic conductor $9.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1176,2,Mod(521,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.u (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.39040727770\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + 1332 x^{7} - 846 x^{6} - 1296 x^{5} + 5265 x^{4} - 10206 x^{3} + 13851 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1097.8
Root \(1.22961 + 1.21986i\) of defining polynomial
Character \(\chi\) \(=\) 1176.1097
Dual form 1176.2.u.b.521.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.67480 - 0.441628i) q^{3} +(-1.40397 + 2.43175i) q^{5} +(2.60993 - 1.47928i) q^{9} +O(q^{10})\) \(q+(1.67480 - 0.441628i) q^{3} +(-1.40397 + 2.43175i) q^{5} +(2.60993 - 1.47928i) q^{9} +(4.74645 - 2.74036i) q^{11} -1.35669i q^{13} +(-1.27745 + 4.69274i) q^{15} +(2.88753 + 5.00135i) q^{17} +(-1.71973 - 0.992889i) q^{19} +(-2.09928 - 1.21202i) q^{23} +(-1.44228 - 2.49811i) q^{25} +(3.71783 - 3.63012i) q^{27} +7.05668i q^{29} +(3.07596 - 1.77591i) q^{31} +(6.73914 - 6.68573i) q^{33} +(-2.14377 + 3.71312i) q^{37} +(-0.599153 - 2.27219i) q^{39} -1.81976 q^{41} +11.2288 q^{43} +(-0.0670332 + 8.42358i) q^{45} +(-0.201213 + 0.348512i) q^{47} +(7.04478 + 7.10106i) q^{51} +(5.28097 - 3.04897i) q^{53} +15.3896i q^{55} +(-3.31870 - 0.903412i) q^{57} +(1.28234 + 2.22108i) q^{59} +(4.75817 + 2.74713i) q^{61} +(3.29914 + 1.90476i) q^{65} +(3.45238 + 5.97970i) q^{67} +(-4.05114 - 1.10279i) q^{69} +2.08251i q^{71} +(0.295696 - 0.170720i) q^{73} +(-3.51878 - 3.54689i) q^{75} +(1.19139 - 2.06355i) q^{79} +(4.62347 - 7.72163i) q^{81} -11.8717 q^{83} -16.2161 q^{85} +(3.11643 + 11.8186i) q^{87} +(-0.576571 + 0.998650i) q^{89} +(4.36734 - 4.33272i) q^{93} +(4.82892 - 2.78798i) q^{95} -16.0187i q^{97} +(8.33413 - 14.1735i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{9} + 8 q^{15} + 6 q^{19} - 18 q^{25} + 48 q^{31} + 12 q^{33} - 2 q^{37} - 22 q^{39} + 20 q^{43} + 42 q^{45} + 6 q^{51} - 8 q^{57} - 36 q^{61} + 14 q^{67} - 30 q^{73} - 54 q^{75} + 28 q^{79} + 30 q^{81} + 16 q^{85} - 78 q^{87} + 16 q^{93} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.67480 0.441628i 0.966948 0.254974i
\(4\) 0 0
\(5\) −1.40397 + 2.43175i −0.627876 + 1.08751i 0.360101 + 0.932913i \(0.382742\pi\)
−0.987977 + 0.154600i \(0.950591\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.60993 1.47928i 0.869977 0.493093i
\(10\) 0 0
\(11\) 4.74645 2.74036i 1.43111 0.826250i 0.433902 0.900960i \(-0.357136\pi\)
0.997205 + 0.0747101i \(0.0238032\pi\)
\(12\) 0 0
\(13\) 1.35669i 0.376279i −0.982142 0.188139i \(-0.939754\pi\)
0.982142 0.188139i \(-0.0602457\pi\)
\(14\) 0 0
\(15\) −1.27745 + 4.69274i −0.329836 + 1.21166i
\(16\) 0 0
\(17\) 2.88753 + 5.00135i 0.700329 + 1.21301i 0.968351 + 0.249593i \(0.0802967\pi\)
−0.268022 + 0.963413i \(0.586370\pi\)
\(18\) 0 0
\(19\) −1.71973 0.992889i −0.394534 0.227784i 0.289589 0.957151i \(-0.406481\pi\)
−0.684123 + 0.729367i \(0.739815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.09928 1.21202i −0.437730 0.252723i 0.264904 0.964275i \(-0.414660\pi\)
−0.702634 + 0.711551i \(0.747993\pi\)
\(24\) 0 0
\(25\) −1.44228 2.49811i −0.288457 0.499622i
\(26\) 0 0
\(27\) 3.71783 3.63012i 0.715497 0.698616i
\(28\) 0 0
\(29\) 7.05668i 1.31039i 0.755458 + 0.655197i \(0.227414\pi\)
−0.755458 + 0.655197i \(0.772586\pi\)
\(30\) 0 0
\(31\) 3.07596 1.77591i 0.552459 0.318962i −0.197654 0.980272i \(-0.563332\pi\)
0.750113 + 0.661309i \(0.229999\pi\)
\(32\) 0 0
\(33\) 6.73914 6.68573i 1.17313 1.16384i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.14377 + 3.71312i −0.352434 + 0.610434i −0.986675 0.162701i \(-0.947979\pi\)
0.634241 + 0.773135i \(0.281313\pi\)
\(38\) 0 0
\(39\) −0.599153 2.27219i −0.0959413 0.363842i
\(40\) 0 0
\(41\) −1.81976 −0.284199 −0.142100 0.989852i \(-0.545385\pi\)
−0.142100 + 0.989852i \(0.545385\pi\)
\(42\) 0 0
\(43\) 11.2288 1.71238 0.856188 0.516665i \(-0.172827\pi\)
0.856188 + 0.516665i \(0.172827\pi\)
\(44\) 0 0
\(45\) −0.0670332 + 8.42358i −0.00999272 + 1.25571i
\(46\) 0 0
\(47\) −0.201213 + 0.348512i −0.0293500 + 0.0508356i −0.880327 0.474367i \(-0.842677\pi\)
0.850977 + 0.525202i \(0.176010\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 7.04478 + 7.10106i 0.986466 + 0.994348i
\(52\) 0 0
\(53\) 5.28097 3.04897i 0.725397 0.418808i −0.0913389 0.995820i \(-0.529115\pi\)
0.816736 + 0.577012i \(0.195781\pi\)
\(54\) 0 0
\(55\) 15.3896i 2.07513i
\(56\) 0 0
\(57\) −3.31870 0.903412i −0.439573 0.119660i
\(58\) 0 0
\(59\) 1.28234 + 2.22108i 0.166947 + 0.289161i 0.937345 0.348403i \(-0.113276\pi\)
−0.770398 + 0.637563i \(0.779942\pi\)
\(60\) 0 0
\(61\) 4.75817 + 2.74713i 0.609222 + 0.351734i 0.772661 0.634819i \(-0.218925\pi\)
−0.163439 + 0.986553i \(0.552259\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.29914 + 1.90476i 0.409208 + 0.236257i
\(66\) 0 0
\(67\) 3.45238 + 5.97970i 0.421775 + 0.730536i 0.996113 0.0880819i \(-0.0280737\pi\)
−0.574338 + 0.818618i \(0.694740\pi\)
\(68\) 0 0
\(69\) −4.05114 1.10279i −0.487700 0.132761i
\(70\) 0 0
\(71\) 2.08251i 0.247148i 0.992335 + 0.123574i \(0.0394357\pi\)
−0.992335 + 0.123574i \(0.960564\pi\)
\(72\) 0 0
\(73\) 0.295696 0.170720i 0.0346086 0.0199813i −0.482596 0.875843i \(-0.660306\pi\)
0.517204 + 0.855862i \(0.326973\pi\)
\(74\) 0 0
\(75\) −3.51878 3.54689i −0.406313 0.409560i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.19139 2.06355i 0.134042 0.232168i −0.791189 0.611572i \(-0.790538\pi\)
0.925231 + 0.379404i \(0.123871\pi\)
\(80\) 0 0
\(81\) 4.62347 7.72163i 0.513719 0.857958i
\(82\) 0 0
\(83\) −11.8717 −1.30309 −0.651543 0.758611i \(-0.725878\pi\)
−0.651543 + 0.758611i \(0.725878\pi\)
\(84\) 0 0
\(85\) −16.2161 −1.75888
\(86\) 0 0
\(87\) 3.11643 + 11.8186i 0.334116 + 1.26708i
\(88\) 0 0
\(89\) −0.576571 + 0.998650i −0.0611164 + 0.105857i −0.894965 0.446137i \(-0.852799\pi\)
0.833848 + 0.551994i \(0.186133\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.36734 4.33272i 0.452872 0.449283i
\(94\) 0 0
\(95\) 4.82892 2.78798i 0.495437 0.286041i
\(96\) 0 0
\(97\) 16.0187i 1.62645i −0.581950 0.813225i \(-0.697710\pi\)
0.581950 0.813225i \(-0.302290\pi\)
\(98\) 0 0
\(99\) 8.33413 14.1735i 0.837612 1.42449i
\(100\) 0 0
\(101\) −7.33982 12.7129i −0.730339 1.26498i −0.956738 0.290950i \(-0.906029\pi\)
0.226399 0.974035i \(-0.427305\pi\)
\(102\) 0 0
\(103\) −4.06960 2.34958i −0.400989 0.231511i 0.285922 0.958253i \(-0.407700\pi\)
−0.686911 + 0.726742i \(0.741034\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.14150 4.12315i −0.690395 0.398600i 0.113365 0.993553i \(-0.463837\pi\)
−0.803760 + 0.594954i \(0.797170\pi\)
\(108\) 0 0
\(109\) −4.41113 7.64030i −0.422509 0.731808i 0.573675 0.819083i \(-0.305517\pi\)
−0.996184 + 0.0872755i \(0.972184\pi\)
\(110\) 0 0
\(111\) −1.95058 + 7.16550i −0.185141 + 0.680119i
\(112\) 0 0
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) 5.89467 3.40329i 0.549680 0.317358i
\(116\) 0 0
\(117\) −2.00693 3.54087i −0.185540 0.327354i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.51916 16.4877i 0.865378 1.49888i
\(122\) 0 0
\(123\) −3.04775 + 0.803658i −0.274806 + 0.0724634i
\(124\) 0 0
\(125\) −5.94002 −0.531291
\(126\) 0 0
\(127\) −6.93769 −0.615620 −0.307810 0.951448i \(-0.599596\pi\)
−0.307810 + 0.951448i \(0.599596\pi\)
\(128\) 0 0
\(129\) 18.8060 4.95895i 1.65578 0.436611i
\(130\) 0 0
\(131\) 0.118734 0.205654i 0.0103739 0.0179680i −0.860792 0.508957i \(-0.830031\pi\)
0.871166 + 0.490989i \(0.163365\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.60782 + 14.1374i 0.310511 + 1.21676i
\(136\) 0 0
\(137\) −9.58873 + 5.53606i −0.819221 + 0.472977i −0.850148 0.526544i \(-0.823487\pi\)
0.0309270 + 0.999522i \(0.490154\pi\)
\(138\) 0 0
\(139\) 1.02466i 0.0869108i 0.999055 + 0.0434554i \(0.0138366\pi\)
−0.999055 + 0.0434554i \(0.986163\pi\)
\(140\) 0 0
\(141\) −0.183080 + 0.672550i −0.0154181 + 0.0566389i
\(142\) 0 0
\(143\) −3.71783 6.43947i −0.310900 0.538495i
\(144\) 0 0
\(145\) −17.1601 9.90740i −1.42507 0.822765i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −19.0549 11.0013i −1.56104 0.901266i −0.997152 0.0754127i \(-0.975973\pi\)
−0.563886 0.825853i \(-0.690694\pi\)
\(150\) 0 0
\(151\) 3.63368 + 6.29371i 0.295704 + 0.512175i 0.975149 0.221552i \(-0.0711123\pi\)
−0.679444 + 0.733727i \(0.737779\pi\)
\(152\) 0 0
\(153\) 14.9346 + 8.78171i 1.20739 + 0.709959i
\(154\) 0 0
\(155\) 9.97331i 0.801075i
\(156\) 0 0
\(157\) −19.6994 + 11.3735i −1.57219 + 0.907702i −0.576285 + 0.817249i \(0.695498\pi\)
−0.995901 + 0.0904525i \(0.971169\pi\)
\(158\) 0 0
\(159\) 7.49807 7.43864i 0.594636 0.589923i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −9.06678 + 15.7041i −0.710165 + 1.23004i 0.254630 + 0.967039i \(0.418046\pi\)
−0.964795 + 0.263003i \(0.915287\pi\)
\(164\) 0 0
\(165\) 6.79646 + 25.7745i 0.529104 + 2.00654i
\(166\) 0 0
\(167\) 24.0942 1.86447 0.932233 0.361858i \(-0.117857\pi\)
0.932233 + 0.361858i \(0.117857\pi\)
\(168\) 0 0
\(169\) 11.1594 0.858414
\(170\) 0 0
\(171\) −5.95715 0.0474059i −0.455554 0.00362522i
\(172\) 0 0
\(173\) −5.18802 + 8.98592i −0.394438 + 0.683187i −0.993029 0.117868i \(-0.962394\pi\)
0.598591 + 0.801055i \(0.295727\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.12856 + 3.15356i 0.235157 + 0.237036i
\(178\) 0 0
\(179\) −11.5922 + 6.69274i −0.866439 + 0.500239i −0.866163 0.499761i \(-0.833421\pi\)
−0.000276030 1.00000i \(0.500088\pi\)
\(180\) 0 0
\(181\) 18.4339i 1.37018i −0.728457 0.685092i \(-0.759762\pi\)
0.728457 0.685092i \(-0.240238\pi\)
\(182\) 0 0
\(183\) 9.18221 + 2.49957i 0.678769 + 0.184773i
\(184\) 0 0
\(185\) −6.01960 10.4263i −0.442570 0.766554i
\(186\) 0 0
\(187\) 27.4110 + 15.8258i 2.00449 + 1.15729i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.59492 + 2.07553i 0.260119 + 0.150180i 0.624389 0.781114i \(-0.285348\pi\)
−0.364270 + 0.931293i \(0.618681\pi\)
\(192\) 0 0
\(193\) −9.75462 16.8955i −0.702153 1.21616i −0.967709 0.252069i \(-0.918889\pi\)
0.265556 0.964095i \(-0.414444\pi\)
\(194\) 0 0
\(195\) 6.36661 + 1.73311i 0.455922 + 0.124110i
\(196\) 0 0
\(197\) 3.80952i 0.271417i −0.990749 0.135709i \(-0.956669\pi\)
0.990749 0.135709i \(-0.0433311\pi\)
\(198\) 0 0
\(199\) 5.30327 3.06185i 0.375939 0.217049i −0.300111 0.953904i \(-0.597024\pi\)
0.676050 + 0.736856i \(0.263690\pi\)
\(200\) 0 0
\(201\) 8.42286 + 8.49015i 0.594103 + 0.598849i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.55490 4.42522i 0.178442 0.309071i
\(206\) 0 0
\(207\) −7.27189 0.0578683i −0.505431 0.00402212i
\(208\) 0 0
\(209\) −10.8835 −0.752827
\(210\) 0 0
\(211\) 2.93058 0.201750 0.100875 0.994899i \(-0.467836\pi\)
0.100875 + 0.994899i \(0.467836\pi\)
\(212\) 0 0
\(213\) 0.919693 + 3.48779i 0.0630163 + 0.238979i
\(214\) 0 0
\(215\) −15.7649 + 27.3057i −1.07516 + 1.86223i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0.419838 0.416510i 0.0283700 0.0281451i
\(220\) 0 0
\(221\) 6.78530 3.91749i 0.456428 0.263519i
\(222\) 0 0
\(223\) 4.61145i 0.308806i −0.988008 0.154403i \(-0.950655\pi\)
0.988008 0.154403i \(-0.0493454\pi\)
\(224\) 0 0
\(225\) −7.45966 4.38635i −0.497311 0.292424i
\(226\) 0 0
\(227\) −8.62344 14.9362i −0.572358 0.991353i −0.996323 0.0856745i \(-0.972695\pi\)
0.423965 0.905678i \(-0.360638\pi\)
\(228\) 0 0
\(229\) 11.5705 + 6.68024i 0.764601 + 0.441443i 0.830945 0.556354i \(-0.187800\pi\)
−0.0663443 + 0.997797i \(0.521134\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.5908 9.00135i −1.02139 0.589698i −0.106882 0.994272i \(-0.534087\pi\)
−0.914505 + 0.404574i \(0.867420\pi\)
\(234\) 0 0
\(235\) −0.564996 0.978602i −0.0368563 0.0638370i
\(236\) 0 0
\(237\) 1.08403 3.98220i 0.0704151 0.258671i
\(238\) 0 0
\(239\) 23.6499i 1.52979i −0.644158 0.764893i \(-0.722792\pi\)
0.644158 0.764893i \(-0.277208\pi\)
\(240\) 0 0
\(241\) −3.53574 + 2.04136i −0.227757 + 0.131496i −0.609537 0.792758i \(-0.708645\pi\)
0.381780 + 0.924253i \(0.375311\pi\)
\(242\) 0 0
\(243\) 4.33332 14.9741i 0.277983 0.960586i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.34705 + 2.33315i −0.0857105 + 0.148455i
\(248\) 0 0
\(249\) −19.8827 + 5.24286i −1.26002 + 0.332253i
\(250\) 0 0
\(251\) −5.78085 −0.364884 −0.182442 0.983217i \(-0.558400\pi\)
−0.182442 + 0.983217i \(0.558400\pi\)
\(252\) 0 0
\(253\) −13.2855 −0.835251
\(254\) 0 0
\(255\) −27.1587 + 7.16146i −1.70075 + 0.448468i
\(256\) 0 0
\(257\) −10.4824 + 18.1560i −0.653871 + 1.13254i 0.328304 + 0.944572i \(0.393523\pi\)
−0.982175 + 0.187966i \(0.939810\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 10.4388 + 18.4175i 0.646145 + 1.14001i
\(262\) 0 0
\(263\) 4.32937 2.49957i 0.266961 0.154130i −0.360545 0.932742i \(-0.617409\pi\)
0.627506 + 0.778612i \(0.284076\pi\)
\(264\) 0 0
\(265\) 17.1227i 1.05184i
\(266\) 0 0
\(267\) −0.524611 + 1.92717i −0.0321057 + 0.117941i
\(268\) 0 0
\(269\) 7.67602 + 13.2953i 0.468015 + 0.810626i 0.999332 0.0365470i \(-0.0116359\pi\)
−0.531317 + 0.847173i \(0.678303\pi\)
\(270\) 0 0
\(271\) −14.4761 8.35779i −0.879362 0.507700i −0.00891391 0.999960i \(-0.502837\pi\)
−0.870448 + 0.492260i \(0.836171\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −13.6914 7.90476i −0.825625 0.476675i
\(276\) 0 0
\(277\) 11.2571 + 19.4979i 0.676376 + 1.17152i 0.976065 + 0.217481i \(0.0697839\pi\)
−0.299689 + 0.954037i \(0.596883\pi\)
\(278\) 0 0
\(279\) 5.40098 9.18520i 0.323348 0.549903i
\(280\) 0 0
\(281\) 18.1134i 1.08055i −0.841488 0.540276i \(-0.818320\pi\)
0.841488 0.540276i \(-0.181680\pi\)
\(282\) 0 0
\(283\) 5.00728 2.89095i 0.297652 0.171849i −0.343736 0.939066i \(-0.611692\pi\)
0.641388 + 0.767217i \(0.278359\pi\)
\(284\) 0 0
\(285\) 6.85625 6.80190i 0.406129 0.402910i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.17567 + 14.1607i −0.480921 + 0.832980i
\(290\) 0 0
\(291\) −7.07428 26.8281i −0.414702 1.57269i
\(292\) 0 0
\(293\) 9.38786 0.548445 0.274222 0.961666i \(-0.411580\pi\)
0.274222 + 0.961666i \(0.411580\pi\)
\(294\) 0 0
\(295\) −7.20151 −0.419288
\(296\) 0 0
\(297\) 7.69864 27.4183i 0.446720 1.59097i
\(298\) 0 0
\(299\) −1.64434 + 2.84808i −0.0950945 + 0.164709i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −17.9071 18.0502i −1.02874 1.03696i
\(304\) 0 0
\(305\) −13.3607 + 7.71380i −0.765031 + 0.441691i
\(306\) 0 0
\(307\) 19.7599i 1.12776i 0.825857 + 0.563880i \(0.190692\pi\)
−0.825857 + 0.563880i \(0.809308\pi\)
\(308\) 0 0
\(309\) −7.85341 2.13784i −0.446765 0.121618i
\(310\) 0 0
\(311\) −10.1911 17.6515i −0.577884 1.00092i −0.995722 0.0924025i \(-0.970545\pi\)
0.417838 0.908522i \(-0.362788\pi\)
\(312\) 0 0
\(313\) −6.19972 3.57941i −0.350429 0.202320i 0.314445 0.949276i \(-0.398182\pi\)
−0.664874 + 0.746955i \(0.731515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.81412 + 5.66618i 0.551216 + 0.318245i 0.749612 0.661877i \(-0.230240\pi\)
−0.198396 + 0.980122i \(0.563573\pi\)
\(318\) 0 0
\(319\) 19.3379 + 33.4942i 1.08271 + 1.87531i
\(320\) 0 0
\(321\) −13.7815 3.75158i −0.769208 0.209393i
\(322\) 0 0
\(323\) 11.4680i 0.638096i
\(324\) 0 0
\(325\) −3.38917 + 1.95674i −0.187997 + 0.108540i
\(326\) 0 0
\(327\) −10.7619 10.8479i −0.595136 0.599891i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.41383 16.3052i 0.517431 0.896216i −0.482364 0.875971i \(-0.660222\pi\)
0.999795 0.0202456i \(-0.00644480\pi\)
\(332\) 0 0
\(333\) −0.102355 + 12.8622i −0.00560903 + 0.704846i
\(334\) 0 0
\(335\) −19.3882 −1.05929
\(336\) 0 0
\(337\) 28.9739 1.57831 0.789156 0.614193i \(-0.210518\pi\)
0.789156 + 0.614193i \(0.210518\pi\)
\(338\) 0 0
\(339\) 1.76651 + 6.69921i 0.0959437 + 0.363851i
\(340\) 0 0
\(341\) 9.73325 16.8585i 0.527085 0.912939i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 8.36942 8.30308i 0.450594 0.447023i
\(346\) 0 0
\(347\) −15.6525 + 9.03697i −0.840270 + 0.485130i −0.857356 0.514724i \(-0.827894\pi\)
0.0170860 + 0.999854i \(0.494561\pi\)
\(348\) 0 0
\(349\) 12.8624i 0.688510i 0.938876 + 0.344255i \(0.111868\pi\)
−0.938876 + 0.344255i \(0.888132\pi\)
\(350\) 0 0
\(351\) −4.92495 5.04395i −0.262875 0.269226i
\(352\) 0 0
\(353\) −13.6386 23.6227i −0.725909 1.25731i −0.958599 0.284760i \(-0.908086\pi\)
0.232690 0.972551i \(-0.425247\pi\)
\(354\) 0 0
\(355\) −5.06415 2.92379i −0.268777 0.155178i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.773273 0.446450i −0.0408118 0.0235627i 0.479455 0.877566i \(-0.340834\pi\)
−0.520267 + 0.854004i \(0.674168\pi\)
\(360\) 0 0
\(361\) −7.52834 13.0395i −0.396229 0.686288i
\(362\) 0 0
\(363\) 8.66131 31.8175i 0.454601 1.66999i
\(364\) 0 0
\(365\) 0.958746i 0.0501831i
\(366\) 0 0
\(367\) 9.57418 5.52765i 0.499768 0.288541i −0.228850 0.973462i \(-0.573496\pi\)
0.728618 + 0.684921i \(0.240163\pi\)
\(368\) 0 0
\(369\) −4.74946 + 2.69194i −0.247247 + 0.140137i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.5503 + 20.0057i −0.598053 + 1.03586i 0.395055 + 0.918657i \(0.370725\pi\)
−0.993108 + 0.117201i \(0.962608\pi\)
\(374\) 0 0
\(375\) −9.94836 + 2.62328i −0.513731 + 0.135465i
\(376\) 0 0
\(377\) 9.57375 0.493073
\(378\) 0 0
\(379\) −23.3938 −1.20166 −0.600830 0.799377i \(-0.705163\pi\)
−0.600830 + 0.799377i \(0.705163\pi\)
\(380\) 0 0
\(381\) −11.6193 + 3.06387i −0.595273 + 0.156967i
\(382\) 0 0
\(383\) 11.5139 19.9426i 0.588331 1.01902i −0.406120 0.913820i \(-0.633119\pi\)
0.994451 0.105200i \(-0.0335482\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 29.3064 16.6105i 1.48973 0.844360i
\(388\) 0 0
\(389\) −5.45545 + 3.14970i −0.276602 + 0.159696i −0.631884 0.775063i \(-0.717718\pi\)
0.355282 + 0.934759i \(0.384385\pi\)
\(390\) 0 0
\(391\) 13.9990i 0.707958i
\(392\) 0 0
\(393\) 0.108034 0.396866i 0.00544960 0.0200192i
\(394\) 0 0
\(395\) 3.34537 + 5.79435i 0.168324 + 0.291545i
\(396\) 0 0
\(397\) −6.27940 3.62541i −0.315154 0.181954i 0.334077 0.942546i \(-0.391576\pi\)
−0.649230 + 0.760592i \(0.724909\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.8188 6.82360i −0.590204 0.340755i 0.174974 0.984573i \(-0.444016\pi\)
−0.765178 + 0.643819i \(0.777349\pi\)
\(402\) 0 0
\(403\) −2.40936 4.17314i −0.120019 0.207879i
\(404\) 0 0
\(405\) 12.2859 + 22.0841i 0.610489 + 1.09737i
\(406\) 0 0
\(407\) 23.4989i 1.16479i
\(408\) 0 0
\(409\) 11.9303 6.88797i 0.589916 0.340588i −0.175148 0.984542i \(-0.556041\pi\)
0.765064 + 0.643954i \(0.222707\pi\)
\(410\) 0 0
\(411\) −13.6144 + 13.5065i −0.671547 + 0.666224i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 16.6675 28.8690i 0.818177 1.41712i
\(416\) 0 0
\(417\) 0.452519 + 1.71611i 0.0221600 + 0.0840382i
\(418\) 0 0
\(419\) −6.94914 −0.339488 −0.169744 0.985488i \(-0.554294\pi\)
−0.169744 + 0.985488i \(0.554294\pi\)
\(420\) 0 0
\(421\) −0.349861 −0.0170512 −0.00852560 0.999964i \(-0.502714\pi\)
−0.00852560 + 0.999964i \(0.502714\pi\)
\(422\) 0 0
\(423\) −0.00960700 + 1.20724i −0.000467108 + 0.0586981i
\(424\) 0 0
\(425\) 8.32928 14.4267i 0.404029 0.699800i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −9.07048 9.14295i −0.437927 0.441426i
\(430\) 0 0
\(431\) −17.4513 + 10.0755i −0.840601 + 0.485321i −0.857468 0.514537i \(-0.827964\pi\)
0.0168676 + 0.999858i \(0.494631\pi\)
\(432\) 0 0
\(433\) 1.42453i 0.0684585i −0.999414 0.0342292i \(-0.989102\pi\)
0.999414 0.0342292i \(-0.0108976\pi\)
\(434\) 0 0
\(435\) −33.1152 9.01456i −1.58775 0.432215i
\(436\) 0 0
\(437\) 2.40680 + 4.16870i 0.115133 + 0.199416i
\(438\) 0 0
\(439\) −1.76541 1.01926i −0.0842583 0.0486465i 0.457279 0.889323i \(-0.348824\pi\)
−0.541537 + 0.840677i \(0.682157\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.1751 + 6.45195i 0.530945 + 0.306541i 0.741401 0.671062i \(-0.234162\pi\)
−0.210456 + 0.977603i \(0.567495\pi\)
\(444\) 0 0
\(445\) −1.61898 2.80416i −0.0767471 0.132930i
\(446\) 0 0
\(447\) −36.7717 10.0099i −1.73924 0.473453i
\(448\) 0 0
\(449\) 2.49432i 0.117714i −0.998266 0.0588572i \(-0.981254\pi\)
0.998266 0.0588572i \(-0.0187457\pi\)
\(450\) 0 0
\(451\) −8.63741 + 4.98681i −0.406720 + 0.234820i
\(452\) 0 0
\(453\) 8.86517 + 8.93600i 0.416522 + 0.419850i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.30952 + 12.6605i −0.341925 + 0.592232i −0.984790 0.173748i \(-0.944412\pi\)
0.642865 + 0.765979i \(0.277746\pi\)
\(458\) 0 0
\(459\) 28.8908 + 8.11209i 1.34851 + 0.378640i
\(460\) 0 0
\(461\) 2.83467 0.132024 0.0660120 0.997819i \(-0.478972\pi\)
0.0660120 + 0.997819i \(0.478972\pi\)
\(462\) 0 0
\(463\) 14.1594 0.658042 0.329021 0.944323i \(-0.393281\pi\)
0.329021 + 0.944323i \(0.393281\pi\)
\(464\) 0 0
\(465\) 4.40449 + 16.7033i 0.204253 + 0.774598i
\(466\) 0 0
\(467\) −4.98809 + 8.63963i −0.230821 + 0.399794i −0.958050 0.286601i \(-0.907475\pi\)
0.727229 + 0.686395i \(0.240808\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −27.9698 + 27.7481i −1.28878 + 1.27857i
\(472\) 0 0
\(473\) 53.2969 30.7710i 2.45059 1.41485i
\(474\) 0 0
\(475\) 5.72811i 0.262824i
\(476\) 0 0
\(477\) 9.27269 15.7696i 0.424567 0.722041i
\(478\) 0 0
\(479\) 21.7575 + 37.6850i 0.994124 + 1.72187i 0.590805 + 0.806815i \(0.298810\pi\)
0.403320 + 0.915059i \(0.367856\pi\)
\(480\) 0 0
\(481\) 5.03757 + 2.90844i 0.229693 + 0.132614i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 38.9535 + 22.4898i 1.76879 + 1.02121i
\(486\) 0 0
\(487\) −18.5796 32.1808i −0.841921 1.45825i −0.888269 0.459324i \(-0.848092\pi\)
0.0463476 0.998925i \(-0.485242\pi\)
\(488\) 0 0
\(489\) −8.24970 + 30.3055i −0.373064 + 1.37046i
\(490\) 0 0
\(491\) 22.1831i 1.00111i 0.865704 + 0.500556i \(0.166871\pi\)
−0.865704 + 0.500556i \(0.833129\pi\)
\(492\) 0 0
\(493\) −35.2929 + 20.3764i −1.58951 + 0.917707i
\(494\) 0 0
\(495\) 22.7655 + 40.1657i 1.02323 + 1.80532i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −8.33695 + 14.4400i −0.373213 + 0.646424i −0.990058 0.140661i \(-0.955077\pi\)
0.616845 + 0.787085i \(0.288411\pi\)
\(500\) 0 0
\(501\) 40.3531 10.6407i 1.80284 0.475390i
\(502\) 0 0
\(503\) −8.55884 −0.381620 −0.190810 0.981627i \(-0.561111\pi\)
−0.190810 + 0.981627i \(0.561111\pi\)
\(504\) 0 0
\(505\) 41.2196 1.83425
\(506\) 0 0
\(507\) 18.6898 4.92829i 0.830042 0.218873i
\(508\) 0 0
\(509\) 14.1072 24.4345i 0.625292 1.08304i −0.363192 0.931714i \(-0.618313\pi\)
0.988484 0.151324i \(-0.0483536\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −9.99798 + 2.55144i −0.441422 + 0.112649i
\(514\) 0 0
\(515\) 11.4272 6.59750i 0.503543 0.290721i
\(516\) 0 0
\(517\) 2.20559i 0.0970017i
\(518\) 0 0
\(519\) −4.72049 + 17.3408i −0.207206 + 0.761177i
\(520\) 0 0
\(521\) 9.00041 + 15.5892i 0.394315 + 0.682974i 0.993014 0.118001i \(-0.0376485\pi\)
−0.598698 + 0.800975i \(0.704315\pi\)
\(522\) 0 0
\(523\) 11.9049 + 6.87332i 0.520567 + 0.300549i 0.737167 0.675711i \(-0.236163\pi\)
−0.216600 + 0.976260i \(0.569497\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.7639 + 10.2560i 0.773806 + 0.446757i
\(528\) 0 0
\(529\) −8.56202 14.8299i −0.372262 0.644776i
\(530\) 0 0
\(531\) 6.63243 + 3.89993i 0.287823 + 0.169243i
\(532\) 0 0
\(533\) 2.46886i 0.106938i
\(534\) 0 0
\(535\) 20.0530 11.5776i 0.866965 0.500542i
\(536\) 0 0
\(537\) −16.4589 + 16.3284i −0.710254 + 0.704624i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −19.6272 + 33.9953i −0.843839 + 1.46157i 0.0427866 + 0.999084i \(0.486376\pi\)
−0.886626 + 0.462488i \(0.846957\pi\)
\(542\) 0 0
\(543\) −8.14094 30.8732i −0.349361 1.32490i
\(544\) 0 0
\(545\) 24.7724 1.06113
\(546\) 0 0
\(547\) −12.4980 −0.534375 −0.267188 0.963645i \(-0.586094\pi\)
−0.267188 + 0.963645i \(0.586094\pi\)
\(548\) 0 0
\(549\) 16.4823 + 0.131163i 0.703446 + 0.00559789i
\(550\) 0 0
\(551\) 7.00651 12.1356i 0.298487 0.516995i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −14.6862 14.8035i −0.623393 0.628374i
\(556\) 0 0
\(557\) −15.4816 + 8.93830i −0.655976 + 0.378728i −0.790742 0.612150i \(-0.790305\pi\)
0.134766 + 0.990877i \(0.456972\pi\)
\(558\) 0 0
\(559\) 15.2340i 0.644331i
\(560\) 0 0
\(561\) 52.8971 + 14.3996i 2.23332 + 0.607950i
\(562\) 0 0
\(563\) −1.36644 2.36674i −0.0575885 0.0997462i 0.835794 0.549043i \(-0.185008\pi\)
−0.893382 + 0.449297i \(0.851674\pi\)
\(564\) 0 0
\(565\) −9.72702 5.61589i −0.409219 0.236262i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.72971 + 0.998650i 0.0725133 + 0.0418656i 0.535818 0.844333i \(-0.320003\pi\)
−0.463305 + 0.886199i \(0.653337\pi\)
\(570\) 0 0
\(571\) 1.00728 + 1.74466i 0.0421534 + 0.0730118i 0.886332 0.463050i \(-0.153245\pi\)
−0.844179 + 0.536061i \(0.819912\pi\)
\(572\) 0 0
\(573\) 6.93739 + 1.88848i 0.289814 + 0.0788926i
\(574\) 0 0
\(575\) 6.99231i 0.291599i
\(576\) 0 0
\(577\) 22.0199 12.7132i 0.916701 0.529258i 0.0341199 0.999418i \(-0.489137\pi\)
0.882581 + 0.470160i \(0.155804\pi\)
\(578\) 0 0
\(579\) −23.7986 23.9887i −0.989036 0.996938i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 16.7106 28.9435i 0.692080 1.19872i
\(584\) 0 0
\(585\) 11.4282 + 0.0909435i 0.472498 + 0.00376005i
\(586\) 0 0
\(587\) 34.4645 1.42250 0.711251 0.702939i \(-0.248129\pi\)
0.711251 + 0.702939i \(0.248129\pi\)
\(588\) 0 0
\(589\) −7.05312 −0.290619
\(590\) 0 0
\(591\) −1.68239 6.38020i −0.0692043 0.262446i
\(592\) 0 0
\(593\) −3.62199 + 6.27347i −0.148737 + 0.257620i −0.930761 0.365628i \(-0.880854\pi\)
0.782024 + 0.623249i \(0.214188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.52974 7.47006i 0.308172 0.305729i
\(598\) 0 0
\(599\) 32.5464 18.7907i 1.32981 0.767766i 0.344540 0.938772i \(-0.388035\pi\)
0.985270 + 0.171005i \(0.0547015\pi\)
\(600\) 0 0
\(601\) 3.78103i 0.154232i 0.997022 + 0.0771158i \(0.0245711\pi\)
−0.997022 + 0.0771158i \(0.975429\pi\)
\(602\) 0 0
\(603\) 17.8561 + 10.4996i 0.727157 + 0.427575i
\(604\) 0 0
\(605\) 26.7293 + 46.2965i 1.08670 + 1.88222i
\(606\) 0 0
\(607\) 24.0353 + 13.8768i 0.975565 + 0.563242i 0.900928 0.433968i \(-0.142887\pi\)
0.0746364 + 0.997211i \(0.476220\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.472823 + 0.272985i 0.0191284 + 0.0110438i
\(612\) 0 0
\(613\) 15.3570 + 26.5991i 0.620264 + 1.07433i 0.989436 + 0.144968i \(0.0463080\pi\)
−0.369172 + 0.929361i \(0.620359\pi\)
\(614\) 0 0
\(615\) 2.32466 8.53968i 0.0937392 0.344353i
\(616\) 0 0
\(617\) 44.3075i 1.78375i 0.452279 + 0.891877i \(0.350611\pi\)
−0.452279 + 0.891877i \(0.649389\pi\)
\(618\) 0 0
\(619\) −27.4026 + 15.8209i −1.10140 + 0.635895i −0.936589 0.350430i \(-0.886035\pi\)
−0.164813 + 0.986325i \(0.552702\pi\)
\(620\) 0 0
\(621\) −12.2045 + 3.11455i −0.489751 + 0.124982i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15.5511 26.9352i 0.622042 1.07741i
\(626\) 0 0
\(627\) −18.2277 + 4.80645i −0.727945 + 0.191951i
\(628\) 0 0
\(629\) −24.7608 −0.987279
\(630\) 0 0
\(631\) −20.7528 −0.826157 −0.413079 0.910695i \(-0.635547\pi\)
−0.413079 + 0.910695i \(0.635547\pi\)
\(632\) 0 0
\(633\) 4.90815 1.29423i 0.195082 0.0514409i
\(634\) 0 0
\(635\) 9.74033 16.8707i 0.386533 0.669495i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.08061 + 5.43520i 0.121867 + 0.215013i
\(640\) 0 0
\(641\) −33.0033 + 19.0545i −1.30355 + 0.752606i −0.981012 0.193949i \(-0.937870\pi\)
−0.322541 + 0.946556i \(0.604537\pi\)
\(642\) 0 0
\(643\) 29.5791i 1.16648i −0.812298 0.583242i \(-0.801784\pi\)
0.812298 0.583242i \(-0.198216\pi\)
\(644\) 0 0
\(645\) −14.3442 + 52.6939i −0.564804 + 2.07482i
\(646\) 0 0
\(647\) 10.5935 + 18.3485i 0.416474 + 0.721354i 0.995582 0.0938966i \(-0.0299323\pi\)
−0.579108 + 0.815251i \(0.696599\pi\)
\(648\) 0 0
\(649\) 12.1731 + 7.02817i 0.477838 + 0.275880i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23.0548 + 13.3107i 0.902204 + 0.520888i 0.877915 0.478817i \(-0.158934\pi\)
0.0242893 + 0.999705i \(0.492268\pi\)
\(654\) 0 0
\(655\) 0.333399 + 0.577465i 0.0130270 + 0.0225634i
\(656\) 0 0
\(657\) 0.519203 0.882984i 0.0202560 0.0344485i
\(658\) 0 0
\(659\) 16.3864i 0.638322i 0.947701 + 0.319161i \(0.103401\pi\)
−0.947701 + 0.319161i \(0.896599\pi\)
\(660\) 0 0
\(661\) 16.0227 9.25072i 0.623211 0.359811i −0.154907 0.987929i \(-0.549508\pi\)
0.778118 + 0.628118i \(0.216174\pi\)
\(662\) 0 0
\(663\) 9.63396 9.55760i 0.374152 0.371186i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.55284 14.8139i 0.331167 0.573598i
\(668\) 0 0
\(669\) −2.03654 7.72327i −0.0787373 0.298599i
\(670\) 0 0
\(671\) 30.1125 1.16248
\(672\) 0 0
\(673\) −45.4357 −1.75142 −0.875708 0.482841i \(-0.839605\pi\)
−0.875708 + 0.482841i \(0.839605\pi\)
\(674\) 0 0
\(675\) −14.4306 4.05189i −0.555434 0.155957i
\(676\) 0 0
\(677\) −15.8566 + 27.4644i −0.609419 + 1.05554i 0.381917 + 0.924196i \(0.375264\pi\)
−0.991336 + 0.131348i \(0.958069\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −21.0388 21.2069i −0.806209 0.812650i
\(682\) 0 0
\(683\) 31.0917 17.9508i 1.18969 0.686868i 0.231454 0.972846i \(-0.425652\pi\)
0.958236 + 0.285978i \(0.0923183\pi\)
\(684\) 0 0
\(685\) 31.0899i 1.18788i
\(686\) 0 0
\(687\) 22.3285 + 6.07823i 0.851886 + 0.231899i
\(688\) 0 0
\(689\) −4.13651 7.16465i −0.157589 0.272952i
\(690\) 0 0
\(691\) −22.2415 12.8411i −0.846106 0.488499i 0.0132293 0.999912i \(-0.495789\pi\)
−0.859335 + 0.511413i \(0.829122\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.49173 1.43860i −0.0945167 0.0545692i
\(696\) 0 0
\(697\) −5.25462 9.10127i −0.199033 0.344735i
\(698\) 0 0
\(699\) −30.0868 8.19016i −1.13799 0.309780i
\(700\) 0 0
\(701\) 1.29881i 0.0490553i −0.999699 0.0245276i \(-0.992192\pi\)
0.999699 0.0245276i \(-0.00780818\pi\)
\(702\) 0 0
\(703\) 7.37344 4.25706i 0.278095 0.160558i
\(704\) 0 0
\(705\) −1.37844 1.38945i −0.0519149 0.0523296i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 13.8609 24.0077i 0.520556 0.901629i −0.479158 0.877728i \(-0.659058\pi\)
0.999714 0.0239010i \(-0.00760863\pi\)
\(710\) 0 0
\(711\) 0.0568835 7.14813i 0.00213330 0.268076i
\(712\) 0 0
\(713\) −8.60973 −0.322437
\(714\) 0 0
\(715\) 20.8789 0.780828
\(716\) 0 0
\(717\) −10.4444 39.6089i −0.390055 1.47922i
\(718\) 0 0
\(719\) 20.9122 36.2210i 0.779893 1.35081i −0.152109 0.988364i \(-0.548607\pi\)
0.932003 0.362451i \(-0.118060\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −5.02015 + 4.98036i −0.186701 + 0.185221i
\(724\) 0 0
\(725\) 17.6284 10.1777i 0.654701 0.377992i
\(726\) 0 0
\(727\) 2.19295i 0.0813319i 0.999173 + 0.0406660i \(0.0129479\pi\)
−0.999173 + 0.0406660i \(0.987052\pi\)
\(728\) 0 0
\(729\) 0.644508 26.9923i 0.0238707 0.999715i
\(730\) 0 0
\(731\) 32.4235 + 56.1592i 1.19923 + 2.07712i
\(732\) 0 0
\(733\) −18.0850 10.4414i −0.667986 0.385662i 0.127327 0.991861i \(-0.459360\pi\)
−0.795313 + 0.606199i \(0.792693\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32.7731 + 18.9215i 1.20721 + 0.696984i
\(738\) 0 0
\(739\) 6.65032 + 11.5187i 0.244636 + 0.423722i 0.962029 0.272947i \(-0.0879982\pi\)
−0.717393 + 0.696668i \(0.754665\pi\)
\(740\) 0 0
\(741\) −1.22565 + 4.50246i −0.0450255 + 0.165402i
\(742\) 0 0
\(743\) 24.8226i 0.910653i −0.890324 0.455327i \(-0.849522\pi\)
0.890324 0.455327i \(-0.150478\pi\)
\(744\) 0 0
\(745\) 53.5051 30.8912i 1.96028 1.13177i
\(746\) 0 0
\(747\) −30.9843 + 17.5615i −1.13366 + 0.642543i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 5.98635 10.3687i 0.218445 0.378358i −0.735888 0.677104i \(-0.763235\pi\)
0.954333 + 0.298746i \(0.0965682\pi\)
\(752\) 0 0
\(753\) −9.68179 + 2.55298i −0.352824 + 0.0930359i
\(754\) 0 0
\(755\) −20.4063 −0.742663
\(756\) 0 0
\(757\) 29.8095 1.08345 0.541723 0.840557i \(-0.317772\pi\)
0.541723 + 0.840557i \(0.317772\pi\)
\(758\) 0 0
\(759\) −22.2506 + 5.86723i −0.807644 + 0.212967i
\(760\) 0 0
\(761\) 16.7439 29.0013i 0.606967 1.05130i −0.384770 0.923012i \(-0.625719\pi\)
0.991737 0.128286i \(-0.0409474\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −42.3228 + 23.9881i −1.53018 + 0.867291i
\(766\) 0 0
\(767\) 3.01333 1.73975i 0.108805 0.0628186i
\(768\) 0 0
\(769\) 19.6491i 0.708566i 0.935138 + 0.354283i \(0.115275\pi\)
−0.935138 + 0.354283i \(0.884725\pi\)
\(770\) 0 0
\(771\) −9.53770 + 35.0370i −0.343492 + 1.26183i
\(772\) 0 0
\(773\) 6.51659 + 11.2871i 0.234385 + 0.405968i 0.959094 0.283088i \(-0.0913589\pi\)
−0.724708 + 0.689056i \(0.758026\pi\)
\(774\) 0 0
\(775\) −8.87282 5.12273i −0.318721 0.184014i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.12951 + 1.80682i 0.112126 + 0.0647362i
\(780\) 0 0
\(781\) 5.70682 + 9.88451i 0.204206 + 0.353695i
\(782\) 0 0
\(783\) 25.6166 + 26.2355i 0.915462 + 0.937582i
\(784\) 0 0
\(785\) 63.8722i 2.27970i
\(786\) 0 0
\(787\) −21.1053 + 12.1852i −0.752324 + 0.434354i −0.826533 0.562888i \(-0.809690\pi\)
0.0742091 + 0.997243i \(0.476357\pi\)
\(788\) 0 0
\(789\) 6.14697 6.09825i 0.218838 0.217103i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.72702 6.45538i 0.132350 0.229237i
\(794\) 0 0
\(795\) 7.56185 + 28.6771i 0.268191 + 1.01707i
\(796\) 0 0
\(797\) −3.14465 −0.111389 −0.0556947 0.998448i \(-0.517737\pi\)
−0.0556947 + 0.998448i \(0.517737\pi\)
\(798\) 0 0
\(799\) −2.32404 −0.0822186
\(800\) 0 0
\(801\) −0.0275286 + 3.45932i −0.000972675 + 0.122229i
\(802\) 0 0
\(803\) 0.935670 1.62063i 0.0330191 0.0571907i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.7274 + 18.8770i 0.659235 + 0.664502i
\(808\) 0 0
\(809\) 5.76799 3.33015i 0.202792 0.117082i −0.395165 0.918610i \(-0.629313\pi\)
0.597957 + 0.801528i \(0.295979\pi\)
\(810\) 0 0
\(811\) 48.8504i 1.71537i 0.514176 + 0.857685i \(0.328098\pi\)
−0.514176 + 0.857685i \(0.671902\pi\)
\(812\) 0 0
\(813\) −27.9357 7.60460i −0.979747 0.266705i
\(814\) 0 0
\(815\) −25.4590 44.0964i −0.891791 1.54463i
\(816\) 0 0
\(817\) −19.3106 11.1490i −0.675591 0.390053i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −35.0636 20.2440i −1.22373 0.706520i −0.258017 0.966140i \(-0.583069\pi\)
−0.965711 + 0.259621i \(0.916402\pi\)
\(822\) 0 0
\(823\) −17.2956 29.9568i −0.602886 1.04423i −0.992382 0.123201i \(-0.960684\pi\)
0.389496 0.921028i \(-0.372649\pi\)
\(824\) 0 0
\(825\) −26.4214 7.19240i −0.919876 0.250407i
\(826\) 0 0
\(827\) 29.3071i 1.01911i −0.860438 0.509555i \(-0.829810\pi\)
0.860438 0.509555i \(-0.170190\pi\)
\(828\) 0 0
\(829\) −12.7957 + 7.38763i −0.444414 + 0.256583i −0.705468 0.708741i \(-0.749263\pi\)
0.261054 + 0.965324i \(0.415930\pi\)
\(830\) 0 0
\(831\) 27.4643 + 27.6837i 0.952727 + 0.960339i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −33.8277 + 58.5912i −1.17065 + 2.02763i
\(836\) 0 0
\(837\) 4.98915 17.7686i 0.172450 0.614173i
\(838\) 0 0
\(839\) −32.0373 −1.10605 −0.553026 0.833164i \(-0.686527\pi\)
−0.553026 + 0.833164i \(0.686527\pi\)
\(840\) 0 0
\(841\) −20.7968 −0.717131
\(842\) 0 0
\(843\) −7.99936 30.3363i −0.275513 1.04484i
\(844\) 0 0
\(845\) −15.6675 + 27.1369i −0.538978 + 0.933537i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.10948 7.05313i 0.243997 0.242063i
\(850\) 0 0
\(851\) 9.00076 5.19659i 0.308542 0.178137i
\(852\) 0 0
\(853\) 33.1110i 1.13370i 0.823821 + 0.566850i \(0.191838\pi\)
−0.823821 + 0.566850i \(0.808162\pi\)
\(854\) 0 0
\(855\) 8.47896 14.4198i 0.289974 0.493145i
\(856\) 0 0
\(857\) 9.00041 + 15.5892i 0.307448 + 0.532516i 0.977803 0.209524i \(-0.0671916\pi\)
−0.670355 + 0.742040i \(0.733858\pi\)
\(858\) 0 0
\(859\) 23.1107 + 13.3430i 0.788528 + 0.455257i 0.839444 0.543446i \(-0.182881\pi\)
−0.0509160 + 0.998703i \(0.516214\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.69289 + 5.01884i 0.295909 + 0.170843i 0.640604 0.767872i \(-0.278684\pi\)
−0.344694 + 0.938715i \(0.612017\pi\)
\(864\) 0 0
\(865\) −14.5677 25.2320i −0.495317 0.857913i
\(866\) 0 0
\(867\) −7.43889 + 27.3269i −0.252638 + 0.928071i
\(868\) 0 0
\(869\) 13.0594i 0.443009i
\(870\) 0 0
\(871\) 8.11262 4.68382i 0.274885 0.158705i
\(872\) 0 0
\(873\) −23.6961 41.8076i −0.801990 1.41497i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.38102 12.7843i 0.249239 0.431695i −0.714076 0.700069i \(-0.753153\pi\)
0.963315 + 0.268373i \(0.0864861\pi\)
\(878\) 0 0
\(879\) 15.7228 4.14594i 0.530318 0.139839i
\(880\) 0 0
\(881\) −4.42345 −0.149030 −0.0745148 0.997220i \(-0.523741\pi\)
−0.0745148 + 0.997220i \(0.523741\pi\)
\(882\) 0 0
\(883\) −10.5403 −0.354711 −0.177355 0.984147i \(-0.556754\pi\)
−0.177355 + 0.984147i \(0.556754\pi\)
\(884\) 0 0
\(885\) −12.0611 + 3.18038i −0.405430 + 0.106907i
\(886\) 0 0
\(887\) −4.92026 + 8.52213i −0.165206 + 0.286145i −0.936728 0.350057i \(-0.886162\pi\)
0.771522 + 0.636202i \(0.219496\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.785012 49.3203i 0.0262989 1.65229i
\(892\) 0 0
\(893\) 0.692067 0.399565i 0.0231591 0.0133709i
\(894\) 0 0
\(895\) 37.5857i 1.25635i
\(896\) 0 0
\(897\) −1.49615 + 5.49615i −0.0499551 + 0.183511i
\(898\) 0 0
\(899\) 12.5320 + 21.7061i 0.417966 + 0.723939i
\(900\) 0 0
\(901\) 30.4979 + 17.6080i 1.01603 + 0.586607i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 44.8268 + 25.8808i 1.49009 + 0.860306i
\(906\) 0 0
\(907\) 4.10609 + 7.11195i 0.136340 + 0.236148i 0.926109 0.377257i \(-0.123133\pi\)
−0.789768 + 0.613405i \(0.789799\pi\)
\(908\) 0 0
\(909\) −37.9624 22.3222i −1.25913 0.740382i
\(910\) 0 0
\(911\) 44.6131i 1.47810i −0.673652 0.739049i \(-0.735275\pi\)
0.673652 0.739049i \(-0.264725\pi\)
\(912\) 0 0
\(913\) −56.3483 + 32.5327i −1.86486 + 1.07668i
\(914\) 0 0
\(915\) −18.9699 + 18.8196i −0.627126 + 0.622155i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −15.3222 + 26.5388i −0.505431 + 0.875433i 0.494549 + 0.869150i \(0.335333\pi\)
−0.999980 + 0.00628290i \(0.998000\pi\)
\(920\) 0 0
\(921\) 8.72654 + 33.0940i 0.287549 + 1.09048i
\(922\) 0 0
\(923\) 2.82532 0.0929966
\(924\) 0 0
\(925\) 12.3677 0.406648
\(926\) 0 0
\(927\) −14.0970 0.112182i −0.463008 0.00368453i
\(928\) 0 0
\(929\) 10.7198 18.5673i 0.351706 0.609173i −0.634842 0.772642i \(-0.718935\pi\)
0.986548 + 0.163469i \(0.0522683\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24.8635 25.0621i −0.813993 0.820496i
\(934\) 0 0
\(935\) −76.9687 + 44.4379i −2.51714 + 1.45327i
\(936\) 0 0
\(937\) 25.1409i 0.821319i −0.911789 0.410659i \(-0.865299\pi\)
0.911789 0.410659i \(-0.134701\pi\)
\(938\) 0 0
\(939\) −11.9641 3.25684i −0.390433 0.106283i
\(940\) 0 0
\(941\) −26.0598 45.1369i −0.849525 1.47142i −0.881633 0.471936i \(-0.843555\pi\)
0.0321082 0.999484i \(-0.489778\pi\)
\(942\) 0 0
\(943\) 3.82019 + 2.20559i 0.124403 + 0.0718238i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 35.8943 + 20.7236i 1.16641 + 0.673427i 0.952832 0.303500i \(-0.0981551\pi\)
0.213578 + 0.976926i \(0.431488\pi\)
\(948\) 0 0
\(949\) −0.231615 0.401169i −0.00751853 0.0130225i
\(950\) 0 0
\(951\) 18.9391 + 5.15556i 0.614141 + 0.167180i
\(952\) 0 0
\(953\) 29.7579i 0.963952i 0.876184 + 0.481976i \(0.160081\pi\)
−0.876184 + 0.481976i \(0.839919\pi\)
\(954\) 0 0
\(955\) −10.0943 + 5.82797i −0.326645 + 0.188589i
\(956\) 0 0
\(957\) 47.1791 + 47.5560i 1.52508 + 1.53727i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9.19231 + 15.9215i −0.296526 + 0.513598i
\(962\) 0 0
\(963\) −24.7381 0.196861i −0.797174 0.00634376i
\(964\) 0 0
\(965\) 54.7809 1.76346
\(966\) 0 0
\(967\) −16.6814 −0.536436 −0.268218 0.963358i \(-0.586435\pi\)
−0.268218 + 0.963358i \(0.586435\pi\)
\(968\) 0 0
\(969\) −5.06458 19.2066i −0.162698 0.617006i
\(970\) 0 0
\(971\) 25.6466 44.4211i 0.823037 1.42554i −0.0803734 0.996765i \(-0.525611\pi\)
0.903410 0.428777i \(-0.141055\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4.81204 + 4.77390i −0.154109 + 0.152887i
\(976\) 0 0
\(977\) −16.6912 + 9.63669i −0.534000 + 0.308305i −0.742644 0.669687i \(-0.766428\pi\)
0.208644 + 0.977992i \(0.433095\pi\)
\(978\) 0 0
\(979\) 6.32005i 0.201990i
\(980\) 0 0
\(981\) −22.8149 13.4154i −0.728422 0.428319i
\(982\) 0 0
\(983\) 1.85925 + 3.22031i 0.0593008 + 0.102712i 0.894152 0.447764i \(-0.147780\pi\)
−0.834851 + 0.550476i \(0.814446\pi\)
\(984\) 0 0
\(985\) 9.26382 + 5.34847i 0.295170 + 0.170416i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −23.5724 13.6095i −0.749558 0.432758i
\(990\) 0 0
\(991\) 25.4914 + 44.1525i 0.809762 + 1.40255i 0.913029 + 0.407896i \(0.133737\pi\)
−0.103266 + 0.994654i \(0.532929\pi\)
\(992\) 0 0
\(993\) 8.56547 31.4654i 0.271817 0.998526i
\(994\) 0 0
\(995\) 17.1950i 0.545118i
\(996\) 0 0
\(997\) −45.6801 + 26.3734i −1.44670 + 0.835254i −0.998283 0.0585692i \(-0.981346\pi\)
−0.448419 + 0.893823i \(0.648013\pi\)
\(998\) 0 0
\(999\) 5.50889 + 21.5869i 0.174294 + 0.682980i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1176.2.u.b.1097.8 16
3.2 odd 2 inner 1176.2.u.b.1097.6 16
7.2 even 3 1176.2.k.a.881.8 16
7.3 odd 6 inner 1176.2.u.b.521.6 16
7.4 even 3 168.2.u.a.17.3 yes 16
7.5 odd 6 1176.2.k.a.881.9 16
7.6 odd 2 168.2.u.a.89.1 yes 16
21.2 odd 6 1176.2.k.a.881.10 16
21.5 even 6 1176.2.k.a.881.7 16
21.11 odd 6 168.2.u.a.17.1 16
21.17 even 6 inner 1176.2.u.b.521.8 16
21.20 even 2 168.2.u.a.89.3 yes 16
28.11 odd 6 336.2.bc.f.17.6 16
28.19 even 6 2352.2.k.i.881.8 16
28.23 odd 6 2352.2.k.i.881.9 16
28.27 even 2 336.2.bc.f.257.8 16
84.11 even 6 336.2.bc.f.17.8 16
84.23 even 6 2352.2.k.i.881.7 16
84.47 odd 6 2352.2.k.i.881.10 16
84.83 odd 2 336.2.bc.f.257.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.u.a.17.1 16 21.11 odd 6
168.2.u.a.17.3 yes 16 7.4 even 3
168.2.u.a.89.1 yes 16 7.6 odd 2
168.2.u.a.89.3 yes 16 21.20 even 2
336.2.bc.f.17.6 16 28.11 odd 6
336.2.bc.f.17.8 16 84.11 even 6
336.2.bc.f.257.6 16 84.83 odd 2
336.2.bc.f.257.8 16 28.27 even 2
1176.2.k.a.881.7 16 21.5 even 6
1176.2.k.a.881.8 16 7.2 even 3
1176.2.k.a.881.9 16 7.5 odd 6
1176.2.k.a.881.10 16 21.2 odd 6
1176.2.u.b.521.6 16 7.3 odd 6 inner
1176.2.u.b.521.8 16 21.17 even 6 inner
1176.2.u.b.1097.6 16 3.2 odd 2 inner
1176.2.u.b.1097.8 16 1.1 even 1 trivial
2352.2.k.i.881.7 16 84.23 even 6
2352.2.k.i.881.8 16 28.19 even 6
2352.2.k.i.881.9 16 28.23 odd 6
2352.2.k.i.881.10 16 84.47 odd 6